Obstructions to Integrability of Nearly Integrable Dynamical Systems Near Regular Level Sets

We study the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij, including non-Hamiltonian ones. In particular, we assume that there exists a family of periodic orbits on a regular level set of the first integrals having a connected and compact component and give sufficient conditions for nonexistence of the same number of real-analytic first integrals in the perturbed systems as the unperturbed ones and for their real-analytic nonintegrability near the level set such that the first integrals and commutative vector fields depend analytically on the small parameter. We compare our results with the classical results of Poincaré and Kozlov for systems written in action and angle coordinates and discuss their relationships with the subharmonic and homoclinic Melnikov methods for periodic perturbations of single-degree-of-freedom Hamiltonian systems. In particular, the latter discussion reveals that the perturbed systems can be real-analytically nonintgrable even if there exists no transverse homoclinic orbit to a periodic orbit. We illustrate our theory with three examples containing the periodically forced Duffing oscillator.